A small snippet to show the difference:

// declare a array variable that can hold a reference.
String [] array;

// make it null, to indicate it does not refer anything.
array = null;

// at this point there is just a array var initialized to null but no actual array.

// now allocate an array.
array = new String[3];

// now make the individual array elements null..although they already are null.
for(int i=0;i<array.length;i++)
{
    array[i] = null;
}
// at this point we have a array variable, holding reference to an array, 
// whose individual elements are null.

No you can't remove an element from an array, as in making it shorter. Java arrays are fixed-size. You need to use an ArrayList for that.

If you set an element to null, the array will still have the same size, but with a null reference at that point.

 // Let's say a = [0,1,2,3,4] (Integer[])
 a[2] = null;
 // Now a = [0,1,null,3,4]

Correct. int is a primitive type, which means that it contains an explicit value (a number from -2^31 to 2^31-1), and not a reference, so it can't be null. If you really need null values, use Integer.

Primitive Java integers cannot be null, but the Integer class, which wraps a primitive int can be null. Here is one option to consider:

Integer[] num = new Integer[3];
num[0] = 23;
num[1] = null;
num[2] = 12;

The boxing rules in Java make it fairly easy to use int and Integer interchangeably, most of the time.